Basically half equations show what is happening to the electrons. The whole point of electrolysis is to split up an ionic compound and turn the ions back into electrons. So you have to add and remove electrons to get atoms.
Half equations show how you get those atoms from the ions.
Say you have NaCl, and so you get an aqeuous solution of Na+ ions and Cl- ions. The Na+ go to the negative electrode (cathode). So Na+ gains an electron to become Na.
The half equation is: Na+ + e- => Na
You also have Cl-. This is attratced to the positive electrode (anode). So Cl- loses an electron to become Cl.